Somashekhar Naimpally
Proximity and Hyperspace Topologies
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 57, 99-110(2003)
MSC:
54E05 Proximity structures and generalizations
54E15 Uniform structures and generalizations
54B20 Hyperspaces
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54D30 Compactness
Abstract: In this paper we give a survey of the use of proximities in
hyperspace topologies. A proximal hypertopology corresponding to a
LO-proximity is a g eneralization of the well known Vietoris
hypertopology equals the Hausdorff uniform topology corresponding
to the totally bounded uniformity and, being contained in both the
Vietoris and Hausdorff uniform topologies, serves as a bridge
between the two. Wattenberg and Beer-Himmelberg-Prickry-Van Vleck
showed that the locally finite hypertopology induced by a
metrizable space is the sup of the Hausdorff metric topologies
induced by all compatible metrics. Naimpally-Sharma showed that
this follows from the fact that a Tychonoff space is normal iff
its fine uniformity induces the locally finite hypertopology. Di
Concilio-Naimpally-Sharma showed that in a Tychonoff space the
fine uniformity induces the proximal locally finite hypertopology.

We study DELTA topologies introduced by Poppe, and their proximal
variations. We show that a short proof can be given of the
Beer-Tamaki result concerning the uniformizability of (proximal)
DELTA hypertopologies via the Attouch-Wets approach used by Beer
in dealing with the Fell topology. Finally we present a result
concerning (Proximal) DELTA-U-hypertopolgies. Several new
hypertopologies are introduced.
Keywords: ap proximity, hyperspace, $\Delta$-topology, proximal $\Delta$-topology, U-topology, $\Delta\text{U}$-topology, proximal $\Delta\text{U}$-topology, Function space, Vietoris topology, Fell topology, Hausdorff uniformity
Notes: Dedicated to my friend Professor Dr. Harry Poppe on his 70$^{th}$ birthday